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Introductory Statistics Explained: Some Fundamental Concepts in Sampling Exercises © 2022, 2023, 2024 Jeremy Balka Chapter 2: Some Fundamental Concepts in Sampling v1.11 W23 Draft J.B.’s strongly suggested exercises: 1 , 2 , 3 , 4 , 5 , 6 , 8 , 9 , 12 , 15 , 17 , 18 , 21 , 24 , 29 , 31 , 32 NB The section titles and numbers are not yet synced up with the text. 1 Populations and Samples, Parameters and Statistics 1. Succinctly state the di ff erence between a parameter and a statistic. 2. Researchers are interested in estimating the average weight of three-year-old walleye in a lake. A netting program yielded 52 three-year-old walleye, with an average weight of 273 grams. (a) What is the population of interest? (b) What is the sample? (c) Is the value 273 a parameter or a statistic? (d) What is the parameter of interest in this scenario? (e) Is it possible to know the value of the parameter of interest in this scenario? 2 Types of Sampling 2.1 Simple Random Sampling 3. An importer of used medical equipment wants to estimate the percentage of defective pacemakers in a shipment of 200 pacemakers. They open the shipping container, remove the top 8 pacemakers, and find that 0% of of these 8 pacemakers are defective. Unknown to the importer, 10% of the pacemakers in the entire shipment are defective. (a) What is the population of interest? (b) What is the sample? (c) What value given above is a statistic ? (d) What value given above is a parameter ? (e) Would this sample above constitute a simple random sample ? 1 3 yr old walleye 52 3ry old walleye statistic mean or yr old walleye no we need to catch them all 200 pacemakers 8 No only top were sampled

4. Suppose a professor wishes to estimate the proportion of female students in a class of 50 students. They could find out the true proportion without too much di ffi culty, but they decide instead to draw a sample of 10 students. Each student’s ID number is put on a marble. The 50 marbles are put into a box, and the box is shaken vigorously. Suppose after the vigorous shaking that the marbles can be considered to be randomly distributed in the box. Without looking at the marbles, the professor reaches into this box and pulls out 10 marbles without replacement. The 10 students with these ID numbers are a sample from the population of 50 students. Which, if any, of the following statements are true? (a) The 50 students represent the population. (b) The sample can be considered to be a simple random sample from the population. (c) Each possible sample of size 10 has the same chance of being selected. (d) The probability any individual in the population will be in the sample is 0.20. 2.2 Other Types of Random Sampling 5. A professor is interested in learning more about her students, and she decides that she is going to draw a sample of 10 students from her population of 50. She knows she should be drawing some sort of random sample, but doesn’t quite know how to go about it. She considers the following two sampling designs. I. Put all the names of students on small pieces of paper, put them into a hat, and mix them up such that the names are randomly distributed. Then she will pick 10 names without replace- ment. II. To ensure equal representation of males and females, she will have separate hats for males and females. In the first hat she will mix up the names of the male students, then draw 5 names without replacement. She will carry out the same procedure with the females, drawing 5 female names. She will then pool these names together into a group of 10. Which of the following statements are true? (a) Sampling method I results a simple random sample. (b) Sampling method II results in a stratified random sample. (c) Both sampling designs are poorly constructed and should never be used. 3 Experiments and Observational Studies 6. Researchers are investigating the relative e ff ectiveness of two procedures that may prolong life in pa- tients with advanced heart disease. Twenty volunteers are randomly assigned to the two procedures, 10 to procedure A and 10 to procedure B . The time until death is recorded. (a) Is this an experiment or an observational study? (b) What are the experimental units? (c) What are the treatments? (d) What is the response variable? (e) If it is found that those patients from procedure B tended to live much longer on average, is it reasonable to conclude that procedure B caused this increase? T T T T T T stratum may remace where pop students observation researcher unit control other variables imagine É p need to compare to other studies no guaranteed causal relationship b c all other variables arent controlled is there a plausible explanation

7. Researchers are interested in the yield of four types of corn. There are 20 plots of land available, and the researchers randomly assign the types of corn to the 20 plots of land (5 plots for each type of corn). After the growing season, the yield (kg) of corn is measured on each of the 20 plots of land. (a) Is this an observational study or an experiment? (b) What are the experimental units? (c) What are the treatments? (d) What is the response variable? 8. Suppose a long-term observational study finds that children who watch an average of more than 50 hours of television a week are much more likely to be arrested by age 18 than those children who watch less television. Based on this study, is it reasonable to conclude that increased television viewing causes an increase in the likelihood of arrest? 4 Chapter Exercises 9. A study 1 investigated fast food consumption among students on a college campus in the southern United States. The researchers recruited students by posting flyers on campus, o ff ering to give participants small giveaways and a chance to win a $100 gift card from the campus bookstore. The 152 student participants completed several surveys, and one of the surveys involved fast food consumption. In the analysis of the results of one of the surveys, the researchers found that male students tended to spend more per month on fast food than female students did. (a) What is the population of interest? (b) Is the sample of students a simple random sample? (c) What types of bias might be present in a study like this? (d) Would it be possible to conduct a simple random sample of students at this university to investigate fast food consumption? 10. A study 2 investigated physical characteristics of two species of lizard ( Phrynocephalus frontalis and P. versicolor ) found in a region of Inner Mongolia. The researchers were mainly interested in the di ff erences between males and females within each species. Researchers captured these lizards by hand or by noose, and measured various physical characteristics. (a) What are the populations of interest? (b) Did the researchers use a simple random sampling design? (c) What types of bias might be present in this study? 11. A study 3 investigated a possible e ff ect of a vitamin C supplement on the frequency and duration of colds. One thousand volunteers were randomly assigned to two groups. One group received a vitamin C supplement of 1g per day, the other group received a placebo. The participants then kept track of variables related to cold frequency and severity for the duration of the study (approximately 2 months). 1 Heidal et al. (2012). Cost and calorie analysis of fast food consumption in college students. Food and Nutrition Sciences , 3:942–946 2 Qu et al. (2011). Sexual dimorphism and female reproduction in two sympatric toad-headed lizards, Phrynocephalus frontalis and P. versicolor (Agamidae). Animal Biology , 61:139–151 3 Anderson, T., Reid, D., and Beaton, G. (1972). Vitamin C and the common cold: a double-blind trial. Canadian Medical Association Journal , 107:503–508 observational b c diff growing environments butif random diff type Espouse to breed ofcom other is Ip msn.mg fmobs study students on caring a iii T.to ti Ima.rst y Yes in they were randomly secreted from the whole population 2 species of lizard no b c now due been selection bias when catching selection bias